# Computing the implied volatility in stochastic volatility models

@article{Berestycki2004ComputingTI, title={Computing the implied volatility in stochastic volatility models}, author={Henri Berestycki and J{\'e}r{\^o}me Busca and Igor Florent}, journal={Communications on Pure and Applied Mathematics}, year={2004}, volume={57} }

The Black-Scholes model [6, 23] has gained wide recognition on financial markets. One of its shortcomings, however, is that it is inconsistent with most observed option prices. Although the model can still be used very efficiently, it has been proposed to relax its assumptions, and, for instance, to consider that the volatility of the underlying asset S is no longer a constant but rather a stochastic process. There are two well-known approaches to achieve this goal. In the first class of models…

## 201 Citations

### Implied volatility asymptotics under affine stochastic volatility models

- Economics
- 2010

This thesis is concerned with the calibration of affine stochastic volatility models with jumps. This class of models encompasses most models used in practice and captures some of the common features…

### The Term Structure of Implied Volatility in Symmetric Models with Applications to Heston

- Mathematics
- 2010

We study the term structure of the implied volatility in the presence of a symmetric smile. Exploiting the result by Tehranchi (2009) that a symmetric smile generated by a continuous martingale…

### Fast Calibration in the Heston Model

- Economics
- 2012

The Heston model is one of the most popular stochastic volatility models for derivatives pricing. The model proposed by Heston (1993) takes into account non-lognormal distribution of the assets…

### Vanilla Option Pricing on Stochastic Volatility market models

- Mathematics
- 2012

We want to discuss the option pricing on stochastic volatility market models, in which we are going to consider a generic function β (νt ) for the drift of volatility process. It is our intention…

### Asymptotic Expansion for the Normal Implied Volatility in Local Volatility Models

- Mathematics, Economics
- 2011

We study the dynamics of the normal implied volatility in a local volatility model, using a small-time expansion in powers of maturity T. At leading order in this expansion, the asymptotics of the…

### Short-time at-the-money skew and rough fractional volatility

- Mathematics, Economics
- 2015

The Black–Scholes implied volatility skew at the money of SPX options is known to obey a power law with respect to the time to maturity. We construct a model of the underlying asset price process…

### Asymptotics for Exponential Levy Processes and Their Volatility Smile: Survey and New Results

- Mathematics
- 2012

Exponential Levy processes can be used to model the evolution of various financial variables such as FX rates, stock prices, and so on. Considerable efforts have been devoted to pricing derivatives…

### Multiscale Stochastic Volatility for Equity, Interest Rate, and Credit Derivatives

- Economics
- 2011

Building upon the ideas introduced in their previous book, Derivatives in Financial Markets with Stochastic Volatility, the authors study the pricing and hedging of financial derivatives under…

### Markov-functional and stochastic volatility modelling

- Mathematics
- 2012

In this thesis, we study two practical problems in applied mathematical fi
nance. The first topic discusses the issue of pricing and hedging Bermudan swaptions
within a one factor Markov-functional…

### The Short-Time Behaviour of VIX Implied Volatilities in a Multifactor Stochastic Volatility Framework

- Mathematics
- 2017

We consider a modelling setup where the VIX index dynamics are explicitly computable as a smooth transformation of a purely diffusive, multidimensional Markov process. The framework is general enough…

## References

SHOWING 1-10 OF 25 REFERENCES

### Pricing with a Smile

- Economics
- 1994

prices as a function of volatility. If an option price is given by the market we can invert this relationship to get the implied volatility. If the model were perfect, this implied value would be the…

### MANAGING SMILE RISK

- Economics
- 2002

Market smiles and skews are usually managed by using local volatility models a la Dupire. We discover that the dynamics of the market smile predicted by local vol models is opposite of observed…

### Implied and local volatilities under stochastic volatility

- Economics, Mathematics
- 2000

For asset prices that follow stochastic-volatility diffusions, we use asymptotic methods to investigate the behavior of the local volatilities and Black–Scholes volatilities implied by option prices,…

### STOCHASTIC IMPLIED TREES: ARBITRAGE PRICING WITH STOCHASTIC TERM AND STRIKE STRUCTURE OF VOLATILITY

- Economics
- 1998

In this paper we present an arbitrage pricing framework for valuing and hedging contingent equity index claims in the presence of a stochastic term and strike structure of volatility. Our approach to…

### Equivalent Black volatilities

- Mathematics
- 1999

We consider European calls and puts on an asset whose forward price F(t) obeys dF(t)=α(t)A(F)dW(t,) under the forward measure. By using singular perturbation techniques, we obtain explicit algebraic…

### Reconstruction of Volatility: Pricing Index Options by the Steepest Descent Approximation

- Economics
- 2002

We propose a formula for calculating the implied volatility of index options based on the volatility skews of the options on the underlying stocks and on a given correlation matrix for the basket.…

### Implied Trinomial Tress of the Volatility Smile

- Economics
- 1996

In options markets where there is a sign$cant or persistent volatility smile, implied tree models can ensure the consistency o f exotic options prices with the market prices o f liquid standard…

### Implied Trinomial Trees of the Volatility Smile

- Economics
- 1996

This material is for your private information, and we are not soliciting any action based upon it. This report is not to be construed as an offer to sell or the solicitation of an offer to buy any…

### Prices of State-Contingent Claims Implicit in Option Prices

- Economics
- 1978

This paper implements the time-state preference model in a multi-period economy, deriving the prices of primitive securities from the prices of call options on aggregate consumption. These prices…